Classification of nonorientable 3-manifolds admitting decompositions into ≤26 coloured tetrahedra

The present paper adopts a computational approach to the study of nonorientable 3-manifolds: in fact, we describe how to create an automatic catalogue of all nonorientable 3-manifolds admitting coloured triangulations with a fixed number of tetrahedra. In particular, the catalogue has been effectively produced and analysed for up to 26 tetrahedra, to reach the complete classification of all involved 3-manifolds. As a consequence, the following summarising result can be stated: THEOREM I. Exactly seven closed connected prime nonorientable 3-manifolds exist, which admit a coloured triangulation consisting of at most 26 tetrahedra. More precisely, they are the four Euclidean nonorientable 3-manifolds, the nontrivial S-2 bundle over S-1, the topological product between the real projective plane RR2 and S-1, and the torus bundle over S-1, with monodromy induced by matrix [(GRAPHICS)].